b^2 - 4ac = (-4)^2 - 4 imes 2 imes (-6) = 16 + 48 = 64
["# Understanding the Quadratic Formula: Proof via (b² - 4ac) = 64", "The quadratic equation is a cornerstone of algebra, widely used in mathematics, physics, engineering, and economics. At its heart lies the discriminant — the expression ( b^2 - 4ac ) — which determines the nature and number of solutions to any quadratic equation of the form:", "[\nax^2 + bx + c = 0\n]", "Ever wondered how this discriminant is calculated and why replacing values like ( b = -4 ), ( a = 1 ), and ( c = -6 ) leads to a meaningful result? Let’s walk through a clear and step-by-step explanation of evaluating ( b^2 - 4ac ) with specific values and why it equals 64, revealing key insights about the roots.", "---", "### The Quadratic Formula: A Quick Recap", "The solutions to the quadratic equation ( ax^2 + bx + c = 0 ) are given by:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "The term under the square root — the discriminant — is:", "[\nD = b^2 - 4ac\n]", "- If ( D > 0 ): two distinct real roots\n- If ( D = 0 ): one real repeated root\n- If ( D < 0 ): two complex roots", "---", "### Step-by-Step: Evaluating ( b^2 - 4ac ) with Specific Values", "Let’s substitute the values ( b = -4 ), ( a = 1 ), and ( c = -6 ) into the discriminant formula:", "[\nb^2 - 4ac = (-4)^2 - 4 \ imes 1 \ imes (-6)\n]", "Now break it down:", "- ( (-4)^2 = 16 )\n- ( 4 \ imes 1 \ imes (-6) = -24 ), but since this term is subtracted, it becomes: ( -(-24) = +48 )", "So:", "[\n(-4)^2 - 4 \ imes 1 \ imes (-6) = 16 + 48 = 64\n]", "Hence:", "[\nb^2 - 4ac = 64\n]", "This result confirms that the discriminant is 64, a perfect square.", "---", "### Why 64 Matters in the Quadratic Formula", "Because 64 is positive and a perfect square (( 64 = 8^2 )), the quadratic equation has two distinct real roots. Using the full quadratic formula:", "[\nx = \frac{-(-4) \pm \sqrt{64}}{2 \ imes 1} = \frac{4 \pm 8}{2}\n]", "This gives:", "- ( x = \frac{4 + 8}{2} = 6 )\n- ( x = \frac{4 - 8}{2} = -2 )", "Two real solutions confirm the discriminant was correctly evaluated and its value decisively influences the solution’s nature.", "---", "### Real-World Applications of the Discriminant", "- Engineering: Analyzing stress points in structures relies on solving quadratic equations; the discriminant reveals stability conditions.\n- Economics: Profit maximization models often use quadratics; discriminant signs indicate feasible solution regions.\n- Physics: Projectile motion equations yield quadratic forms — real solutions mean a reachable target; complex roots mean no real intersection.", "Understanding how expressions like ( b^2 - 4ac = 64 ) emerge and what they imply empowers precise analysis and accurate interpretations.", "---", "### Summary", "- The equation ( b^2 - 4ac = (-4)^2 - 4 \ imes 1 \ imes (-6) ) simplifies neatly to ( 16 + 48 = 64 ).\n- A positive, perfect square discriminant means two distinct real roots.\n- Mastering discriminants enhances problem-solving across science and engineering disciplines.", "---", "Key Takeaway: Evaluating the discriminant precisely unlocks deep insights about quadratic equations — vital for accurate mathematical modeling and real-world applications.", "---", "### Further Reading", "- Overview of the Quadratic Formula and its Historical Development\n- How to Classify Roots Using the Discriminant\n- Applications of Quadratics in Physics and Finance", "---", "Keywords: ( b^2 - 4ac ), discriminant, quadratic formula, real roots, perfect square, algebra explained, quadratic equations."]
